Why most imperfect information games usually use non machine learning AI?
A heuristic search using MCTS + minimax + alphabeta pruning is a highly efficient AI planning process. What the AI techniques of reinforcement learning (RL) plus neural networks (NNs) typically add to this is a way to establish better heuristics.


  My intuition tells me that this is way harder and far more complex. 


It's not actually that much more complex in concept. Replace the hand-coded heuristic with a learning engine, e.g. DQN or AC3. Train the learning engine from human expert play examples and/or from self play.

It is harder though, because there are many things that can go wrong with an NN-based estimator in a RL context. You will need to make many experiments with different hyper-parameters of the learning engine. For complex games, you may have to invest many 100s of hours of training, which you might want to compare against the end result of a similar amount of time spent refining expert heuristic systems.

For imperfect information games, you may also want to use something that can learn an internal state representation. That could be some kind of explicit belief state that you maintain like an expert system, or something that attempts to learn a good representation, such as an RNN (e.g. LSTM). This may not be necessary for a first try at an agent though, since the MCTS search will make up for some inadequacies of low accuracy heuristics.


  Is that the only reason the NN method is not used?


Up until quite recently, approaches using RL and NN were far harder to find examples of outside of academic machine learning research, and there were not any pre-written frameworks for LSTM or e.g. AC3. In the last few years, RL and NN frameworks have started to appear making an AI self-learning approach far more approachable. 

I would expect that many hobby-level coders considering game-playing AI nowadays would seriously take a look at RL and NNs in order to learn robust heuristics for their game projects. However, the "traditional" search-based methods still work in conjunction with these for a completed agent. 


  Has there been any research to prove how much efficiency edge would be between those 2 approaches?


For card games, I am not aware of any specific research, although I am just a hobbyist, yet to write any specific game engine more complex than tic-tac-toe.

For perfect information board games, the chess playing variant of AlphaZero demonstrates applicability of RL+NN self-play approach versus "traditional" heuristics plus search (represented by Stockfish). However, the framing of the tournament has been criticised as unfair to Stockfish, so it is not necessarily an open-and-shut case that RL is strictly better.